# New Directions in Financial Mathematics and Mathematical Economics (14w5168)

## Organizers

Ulrich Horst (Humboldt University)

Sebastian Jaimungal (University of Toronto)

Ronnie Sircar (Princeton University)

Thaleia Zariphopoulou (University of Texas at Austin)

## Objectives

Analysis of modern equity, energy and commodities markets has proved itself to be a significant driver of mathematical innovation, especially in the areas of stochastic analysis, PDE-theory and convex analysis. For instance, systems of forward-backward stochastic differential equations (FBSDEs) are tailor-made to analyze utility optimization, equilibrium and optimal contracting problems in incomplete markets as well as mean-field games; singular BSDE and PDE systems arise naturally in the microstructure models of emissions trading and in models of optimal portfolio liquidation; principal-agent games leading to variational inequalities arise naturally in models of optimal risk sharing under limited liquidity (operation of large oil rigs or power plants) and in models of delegated portfolio management; problems of natural resource management and exploitation of renewable resources under strategic interaction call for an in-depth analysis of stochastic games based on convex analysis and PDE methods.

Our objective is to discuss recent mathematical developments and innovations arising from the analysis of equity, energy and commodities market. The focus shall be on the following areas:

• Stochastic Control for Portfolio Optimization

• Optimization under Market Impact

• Equilibrium Theory and Dynamic Games

• Mathematics of Energy Markets and Natural Resources.

1.1 Portfolio Optimization

Expected utility models are the cornerstones of theoretical portfolio choice in financial economics. Using the powerful duality technique, the associated stochastic optimization problems have been solved for arbitrary utility functions and general price processes. Recent applications, however, necessitate the development of new models and, in turn, novel techniques and solution approaches.

The modeling, role and effects of multi-scale phenomena in valuation and hedging policies have been extensively examined in the context of stochastic volatility in derivative pricing and credit risk (Fouque, Papanicolaou, Sircar & Sølna 2011). However, the effect of different scales on preferences and/or market input has not been explored for investment models. On the other hand, empirical evidence shows that investors react slower than expected to market changes, hinting that their preferences evolve with time at a different, slower scale (Kahneman 2011). Respectively, multi-scale phenomena for the so-called predictability factors in stock prices have not been analyzed. These phenomena give rise to a family of new interesting homogenization problems for (possible degenerate) Hamilton-Jacobi-Bellman equations, and provide an intriguing link between portfolio sensitivities and derivatives greeks (Fouque, Sircar & Zariphopoulou 2012).

Utilities set at a given time in the future are not rich enough for investment choice with rolling and flexible horizons. A complementary approach is to allow for utilities that are defined for all times and evolve forward in time (Musiela & Zariphopoulou 2010, Henderson & Hobson 2007). Such preferences include all classical ones, and offer a richer framework for preferences, market input and their interdependence. They also allow for a richer framework in risk transfer models and in indifference valuation. They give rise to a family of ill-posed problems which are technically very challenging.

1.2 Optimization under Market Impact

Today most trading in equity markets takes on competing electronic platforms, typically run by exchanges or large brokers. Limit order books give market participants access to buy or sell offers for varying prices and quantities. Traders can choose to add to this collection by a limit (passive) order of their own or to take away from this collection by submitting a market (active) order.

While standard financial market models assume that asset prices follow an exogenous stochastic process, and that all transactions can be settled at the prevailing price without any impact on market dynamics, large market orders tend to have an impact on prices and incoming order flows. Analysis of optimal order placement strategies under market impact has developed into one of the most vibrant area of research in financial mathematics over the last years. Models of optimal portfolio liquidation using active orders have been analyzed using methods from probability and PDE theory. They often give rise to nonlinear PDEs with singular terminal condition (the singularity coming from the liquidation constraint) that cannot be solved using standard methods.

The mathematical analysis becomes even more challenging in models of passive order placements, either because investors place limit orders (Cartea & Jaimungal 2012, Bayraktar & Ludkovski 2011, Bouchard, Dang & Lehalle 2011), or because they are allowed to trade simultaneously in traditional exchanges and alternative trading venues such as such as dark pools (Horst & Naujokat 2011, Kratz & Schoeneborn 2012, Gatheral & Schied 2012). Dark pools are trading venues that allow investors to shield their trading intentions from public view and are executed only if matching liquidity becomes available. Understanding how dark pools and regular markets interact and how to optimally place orders across both markets leads to novel stochastic optimization problems and is an area of much current research. Moreover, allowing agents to place both market and limit orders in an optimal manner result in solving combined singular and impulse control problems and lead to novel and difficult to solve quasi-variational inequalities (Guilbaud & Pham 2012, Cartea & Jaimungal 2012).

1.3 Equilibrium Theory and Dynamic Games

Multi-agent models, in general, describe the interaction between several economic agents, whose individual behavior is based on an optimal portfolio-selection procedure, and its effect on the entire market environment. The most prominent examples are so-called stochastic equilibrium models. Their main objective is to understand the nature of asset price dynamics which would, if consistent with the agent’s rational expectations, lead to market clearing. While being a natural extension of the neoclassical economic paradigm to the financial setting, these models stress the role played by risk and the information flow. In addition to a central place they play in our understanding of the role and the behavior of financial markets, stochastic equilibrium models come with a plethora of interesting mathematical problems on the intersection of optimal stochastic control, convex and functional analysis and the study of systems of partial differential equations and FBSDEs.

The need for better equilibrium models of incomplete markets is by now widely recognized and several innovative approaches have been suggested in recent years (Jofre, Rockafellar & Wets 2010, Filipovic & Kupper 2008). Initially motivated by specific economic questions such as pricing of emissions credits (Carmona, Fehr, Hinz & Porchet 2010) or weather derivatives (Horst, Hu, Imkeller, Reveillac & Zhang 2011), many of these approaches lead to new mathematical results that have applications in other areas of economics such as stochastic and Principal-agents games. Principal-agent games provide a powerful framework of modeling interactions between asymmetrically informed parties including optimal contracting problems under limited liability in the presence of large risks (operation of oil rigs, copper mines, power plants), the analysis of limit order book (Biais, Martimort & Rochet 2000) and the impact of dark pools on traditional equity trading venues; see (Duffie, Malamud & Manso 2009) for an alternative equilibrium approach to dark mar- kets. Principal-agent games are often analyzed using variational analysis methods, hybrid control or complex FBSDE systems (Cvitanic & Zhang 2012).

1.4 Mathematics of Energy Markets and Natural Resources

Commodities and energy markets have gained increased importance both in terms of regulatory concerns such as a way to control emissions (Carmona, Fehr, Hinz & Porchet 2010), and as a growing arena for investment to diversify from at or declining traditional assets. Understanding the increased “financialization” of commodities markets (Tang & Xiong 2009) leads to feedback models, the solution of which poses interesting challenges in nonlinear PDE. Discerning the relative interplay between growing demand due to industrialization and demand due to financial speculation is a delicate inverse problem that requires sophisticated numerical schemes.

Many of these markets are oligopolies, or governed by a few energy providers, and this leads to problems in dynamic game theory that require analysis of BSDEs or systems of PDE (Harris, Howison & Sircar 2010). The resulting game effects build new links between stochastic control and competitive equilibrium. These models pose new challenges in the theory of coupled systems of nonlinear ODEs and PDEs, while giving insight into how to promote the switch from traditional exhaustible energy source to sustainable alternatives, given the ongoing technological advances (Ledvina & Sircar 2012). Extending these frameworks to large player limits and other equilibrium solution concepts connects to the emerging area of mean-field games (Gu ́eant, Lasry & Lions 2010).

Market-based approaches to curbing carbon emissions are unavoidable and require careful design and mathematical analysis of their economic equilibrium effects. This is a new area that researchers in mathematical finance have entered recently (Carmona, Fehr, Hinz & Porchet 2010, Barrieu & Fehr 2011, Belaouar, Fahim & Touzi 2011, Carmona, Delarue, Espinosa & Touzi 2010, Cetin & Verschuere 2009), and the interaction with equilibrium economists, who have not yet embraced the importance of these problems in the mathematics of climate change, will be valuable in both directions. The development of respective analytical and computational tools is crucial in helping energy policy makers trust and rely on quantitative methods in decision making and regulatory functions.

1.5 Conclusion

As evidenced by the many support mails, the proposed workshop equally appeals to mathematicians at the forefront of research in stochastic analysis, optimization and convex analysis, applied mathematicians focusing on transferring mathematical results to other disciplines - especially economics and finance - and mathematical economists specializing in rigorous modeling of real-world phenomena.

The workshop topic is currently of high interest; equity, energy and commodities markets are most major economic importance to many countries including Canada, Mexico and the US. Re- search in the areas of optimization and equilibrium with applications in these areas is taking off. Bringing together mathematicians and mathematical economists at all levels of seniority at the Banff International Research Station will help to further mathematical innovation as well as the transfer of mathematical know-how to key areas of microeconomic theory.

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